Bottom Line:
A player adopting zero-determinant strategies is able to pin the expected payoff of the opponents or to enforce a linear relationship between his own payoff and the opponents' payoff, in a unilateral way.This paper considers zero-determinant strategies in the iterated public goods game, a representative multi-player game where in each round each player will choose whether or not to put his tokens into a public pot, and the tokens in this pot are multiplied by a factor larger than one and then evenly divided among all players.The analytical and numerical results exhibit a similar yet different scenario to the case of two-player games: (i) with small number of players or a small multiplication factor, a player is able to unilaterally pin the expected total payoff of all other players; (ii) a player is able to set the ratio between his payoff and the total payoff of all other players, but this ratio is limited by an upper bound if the multiplication factor exceeds a threshold that depends on the number of players.

ABSTRACTRecently, Press and Dyson have proposed a new class of probabilistic and conditional strategies for the two-player iterated Prisoner's Dilemma, so-called zero-determinant strategies. A player adopting zero-determinant strategies is able to pin the expected payoff of the opponents or to enforce a linear relationship between his own payoff and the opponents' payoff, in a unilateral way. This paper considers zero-determinant strategies in the iterated public goods game, a representative multi-player game where in each round each player will choose whether or not to put his tokens into a public pot, and the tokens in this pot are multiplied by a factor larger than one and then evenly divided among all players. The analytical and numerical results exhibit a similar yet different scenario to the case of two-player games: (i) with small number of players or a small multiplication factor, a player is able to unilaterally pin the expected total payoff of all other players; (ii) a player is able to set the ratio between his payoff and the total payoff of all other players, but this ratio is limited by an upper bound if the multiplication factor exceeds a threshold that depends on the number of players.

f2: The payoff of player 1 versus the average payoff of other two players in a three-player IPGG with r = 1.6. The game is simulated 50000 times and each payoff pair is depicted as a single point in the two-dimensional area. (a) Player 1 adopts a non-ZD strategy with p1 = [1, 0, 0, 0, 0, 1, 1, 1] for the outcomes of {CCC,CCD,CDC,CDD,DCC,DCD,DDC,DDD}, where the payoff pairs are distributed into a two-dimensional area. (b) Player adopts an equalizer strategy p1 = [0.08, 0.15, 0.15, 0.22, 0.17, 0.24, 0.24, 0.31] and player 2 and player 3 both adopt random strategies. The sample points of payoffs form a straight line with slope zero, regardless of player 2’s and player 3’s strategies.(c) Player 1 adopts a χ-extortion strategy with p1 = [0.87, 0.87, 0.87, 0.86, 0.01, 0, 0, 0] and χ = 7.9. The sample points of payoff pairs fall into a straight line with slope less than 1, which indicates the extortioner always seize a larger payoff than the opponents’ average payoff.

Mentions:
where denotes the relation between pC,N−1 and pD,0. The opponents’ total payoff thus depends on the number of players N, the multiplication factor r and the parameter γ. Player 1 can thus adjust the opponents’ total payoff by adopting strategies that results in different values of γ. Note that the same equalizer effect can be realized by different equalizer strategies with the same γ. Figure 2 shows the relationship between player 1’s payoff and the other two players’ average payoff in a three-player IPGG, when player 1 adopts non-ZD and ZD strategies while his opponents adopt random strategies. Under different equalizer strategies, the average payoff of the opponents varies. By inspection on equation (10), a large pC,N−1 or a small pD,0 brings a small γ, and consequently increases the total payoff of the opponents. The range of possible total payoff of the opponents is also strongly affected by r and N: (i) when , player can set this value from (N − 1) to r(N − 1), or equivalently, he can set the average payoff of co-players from 1 to r; (ii) when , the feasible region shrinks as the increase of r; and (iii) when , player can only fix the opponents’ total payoff to (see more detail in Supplementary Methods).

f2: The payoff of player 1 versus the average payoff of other two players in a three-player IPGG with r = 1.6. The game is simulated 50000 times and each payoff pair is depicted as a single point in the two-dimensional area. (a) Player 1 adopts a non-ZD strategy with p1 = [1, 0, 0, 0, 0, 1, 1, 1] for the outcomes of {CCC,CCD,CDC,CDD,DCC,DCD,DDC,DDD}, where the payoff pairs are distributed into a two-dimensional area. (b) Player adopts an equalizer strategy p1 = [0.08, 0.15, 0.15, 0.22, 0.17, 0.24, 0.24, 0.31] and player 2 and player 3 both adopt random strategies. The sample points of payoffs form a straight line with slope zero, regardless of player 2’s and player 3’s strategies.(c) Player 1 adopts a χ-extortion strategy with p1 = [0.87, 0.87, 0.87, 0.86, 0.01, 0, 0, 0] and χ = 7.9. The sample points of payoff pairs fall into a straight line with slope less than 1, which indicates the extortioner always seize a larger payoff than the opponents’ average payoff.

Mentions:
where denotes the relation between pC,N−1 and pD,0. The opponents’ total payoff thus depends on the number of players N, the multiplication factor r and the parameter γ. Player 1 can thus adjust the opponents’ total payoff by adopting strategies that results in different values of γ. Note that the same equalizer effect can be realized by different equalizer strategies with the same γ. Figure 2 shows the relationship between player 1’s payoff and the other two players’ average payoff in a three-player IPGG, when player 1 adopts non-ZD and ZD strategies while his opponents adopt random strategies. Under different equalizer strategies, the average payoff of the opponents varies. By inspection on equation (10), a large pC,N−1 or a small pD,0 brings a small γ, and consequently increases the total payoff of the opponents. The range of possible total payoff of the opponents is also strongly affected by r and N: (i) when , player can set this value from (N − 1) to r(N − 1), or equivalently, he can set the average payoff of co-players from 1 to r; (ii) when , the feasible region shrinks as the increase of r; and (iii) when , player can only fix the opponents’ total payoff to (see more detail in Supplementary Methods).

Bottom Line:
A player adopting zero-determinant strategies is able to pin the expected payoff of the opponents or to enforce a linear relationship between his own payoff and the opponents' payoff, in a unilateral way.This paper considers zero-determinant strategies in the iterated public goods game, a representative multi-player game where in each round each player will choose whether or not to put his tokens into a public pot, and the tokens in this pot are multiplied by a factor larger than one and then evenly divided among all players.The analytical and numerical results exhibit a similar yet different scenario to the case of two-player games: (i) with small number of players or a small multiplication factor, a player is able to unilaterally pin the expected total payoff of all other players; (ii) a player is able to set the ratio between his payoff and the total payoff of all other players, but this ratio is limited by an upper bound if the multiplication factor exceeds a threshold that depends on the number of players.

ABSTRACTRecently, Press and Dyson have proposed a new class of probabilistic and conditional strategies for the two-player iterated Prisoner's Dilemma, so-called zero-determinant strategies. A player adopting zero-determinant strategies is able to pin the expected payoff of the opponents or to enforce a linear relationship between his own payoff and the opponents' payoff, in a unilateral way. This paper considers zero-determinant strategies in the iterated public goods game, a representative multi-player game where in each round each player will choose whether or not to put his tokens into a public pot, and the tokens in this pot are multiplied by a factor larger than one and then evenly divided among all players. The analytical and numerical results exhibit a similar yet different scenario to the case of two-player games: (i) with small number of players or a small multiplication factor, a player is able to unilaterally pin the expected total payoff of all other players; (ii) a player is able to set the ratio between his payoff and the total payoff of all other players, but this ratio is limited by an upper bound if the multiplication factor exceeds a threshold that depends on the number of players.